A fraction is greater than its reciprocal by 9/20. What is the value of this fraction?

Introduction / Context:
This question uses a relationship between a fraction and its reciprocal. Problems of this type commonly appear in algebra sections of aptitude tests. They test your ability to form linear or quadratic equations from verbal statements and then solve them accurately. Here, the fraction is greater than its reciprocal by a fixed amount, 9/20.

Given Data / Assumptions:

• Let the fraction be x (x ≠ 0).
• Its reciprocal is 1/x.
• The fraction is greater than its reciprocal by 9/20, so x - (1/x) = 9/20.
• We need to find the value of x from the given options.

Concept / Approach:
The verbal statement is converted into an algebraic equation involving x and 1/x. To eliminate the denominator, we multiply the entire equation by x, giving a quadratic equation. Quadratic equations of the form a*x^2 + b*x + c = 0 can be solved using the quadratic formula. Finally, we match the valid root with the options provided, typically choosing the positive fraction that fits the context.

Step-by-Step Solution:
Step 1: Write the equation based on the statement: x - 1/x = 9/20. Step 2: Multiply both sides by x to remove the denominator: x * x - 1 = (9/20) * x. Step 3: Simplify to get: x^2 - 1 = (9/20)x. Step 4: Multiply through by 20 to clear the fraction: 20x^2 - 20 = 9x. Step 5: Rearrange into standard quadratic form: 20x^2 - 9x - 20 = 0. Step 6: Use the quadratic formula: x = [9 ± sqrt(9^2 - 4 * 20 * (-20))] / (2 * 20). Step 7: Compute the discriminant: 9^2 = 81; 4 * 20 * 20 = 1600; so 81 + 1600 = 1681. Step 8: The square root of 1681 is 41, so x = (9 ± 41) / 40. Step 9: The roots are x = (9 + 41) / 40 = 50/40 = 5/4 and x = (9 - 41) / 40 = -32/40 = -4/5.

Verification / Alternative Check:
Test x = 5/4 in the original equation: x - 1/x = 5/4 - 4/5. Compute 5/4 - 4/5 = (25/20) - (16/20) = 9/20, which matches the condition. The second root, -4/5, also satisfies the equation algebraically but is negative. The options provided include 5/4 and 4/5, not -4/5, so we select the positive fraction 5/4.

Why Other Options Are Wrong:
4/5 would give 4/5 - 5/4, which is negative, not 9/20.
3/4 and 4/3 also do not satisfy the given equation when substituted into x - 1/x. Their differences from their reciprocals will be different values, not 9/20.

Common Pitfalls:
Students sometimes forget that both positive and negative roots can arise when solving quadratics. Another mistake is to invert the relationship and write 1/x - x instead of x - 1/x. Always read the phrase carefully: “fraction is greater than its reciprocal by” means fraction minus reciprocal equals the given positive difference.

Final Answer:
The required fraction is 5/4.